1.a)Determine dy/dx if y= sqrt[(3x^2+2)^3]

For this question, I got 9/2x(3x^2+2)^1/2 as my answer.

b)State any values of x for which the function is not differentiable

y= sqrt[(3x^2+2)^3]

= (3x^2 + 2)^(3/2)
dy/dx = (3/2)(3x^2 + 2)^(1/2) (6x)
= 9x(3x^2+2)^(1/2)
or
= 9√(3x^2 + 2) ,

which is valid for all values of x
since (3x^2 + 2) is always positive.

To determine dy/dx, you can apply the chain rule and power rule for differentiation. The given function is y = sqrt[(3x^2+2)^3].

First, let's simplify the function before finding its derivative.

Step 1: Apply the power rule to the inside function (3x^2 + 2) by multiplying the exponent (3) to each exponent (2):
y = sqrt[(3x^2+2)^3]
= [(3x^2+2)^3]^1/2
= (3x^2+2)^(3/2).

Now we can find the derivative(dy/dx) using the chain rule and power rule.

Step 2: Apply the chain rule:
dy/dx = (3/2) * (3x^2+2)^(1/2-1) * (6x)
= (9x)(3x^2+2)^(-1/2).

So, the derivative of y = sqrt[(3x^2+2)^3] is dy/dx = (9x)(3x^2+2)^(-1/2).
Therefore, your answer of 9/2x(3x^2+2)^1/2 is not correct.

Moving on to part b) - to find the values of x for which the given function is not differentiable, we need to look for any potential points where the derivative might not exist. In this case, that would be when the denominator of the derivative expression becomes zero.

Since the derivative (9x)(3x^2+2)^(-1/2) involves the expression (3x^2+2) raised to the power -1/2, we need to find the values of x for which (3x^2+2) = 0.

Solving for x in (3x^2+2) = 0:
3x^2 = -2
x^2 = -2/3.

Since the equation x^2 = -2/3 has no real solutions (because the square of a real number cannot be negative), there are no values of x for which the function y = sqrt[(3x^2+2)^3] is not differentiable.

Hence, the function is differentiable for all real values of x.